Question

Given the sequence 5; 12 ; 21; 32; .... a determine the formular for the nth term of thw sequance . 2 determine between which two consecutive terms in the sequance the difference will equal 245 sketch the graph to represent the second difference

Answer 1

First difference
12-5 = 7
21-12 = 9
32-21 = 11

Second difference
9-11 = 2
11-9 = 2

second difference is constant so it could defined by quadratic formula,
u(x) = ax^2 + bx + c
because 2a = Second difference = 2
so a = 1
then u(x) = x^2 + bx + c

build equations from known values to find b and c
for x=1, u(1) = 5 = 1^2 + b(1) + c
b + c = 4 ...(1)

x=2, u(2) = 12 = 2^2 + b(2) + c
2b + c = 8 ... (2)

solve b = 4, c =0
so u(x) = x^2 + 4x ... formula for n term (i use x not n here)

difference between consecutive term is 245, we have
245 = u(x+1) - u(x)
245 = (x+1)^2 + 4(x+1) - (x^2 + 4x)
245 = x^2 + 2x + 1 + 4x + 4 - x^2 - 4x
245 = 2x + 5
x = 120
it's between term 120 and 121

graph is y=2